S'mon said:
In reality, yes, since as Einstein demonstrated, time is relative, not a Newtonian Universal Constant. However Trek writers like to use words like 'quantum' without having any idea what they're talking about. They seemed to assume some absolute frame of temporal reference against which the parts were -403 (etc) years old.
Or you could look at it this way.
Accepted Scientific Fact#1: The universe is expanding all the time.
Accepted Scientific Fact#2: Matter cannot be created or destroyed, only rearanged.
Therefore, the density of the universe becomes less as the Universe expands, meaning the farther back in time, the more densely packed the matter in the universe, and the farther forward, the LESS densely packed.
Possible Trek Science conclusion: The density of the universe when something is created affects its molecular structure, and this 'device' reads the molecular structure to find out the density of the universe, then extrapolates the 'temporal date' of the object it reads and displays it in relation to the current temporal date.
The only thing I haven't figured out is wouldn't that mean that everythings molecular structure is undergoign 'constant' change?
So there's still a few bugs, but if you ignore the little inconsistencies (Which is what you're supposed to do in a science FICTION show) it works.
And you know the weird part? According to advanced Quantum Theory, that's possible... Not with our current technology, but it IS theoretically possible.
Which could explain why they call it 'Quantum' Dating.